This paper studies stochastic algorithms for minimizing paraconvex functions, a function class that generalizes weakly convex functions and includes, for instance, H\"older smooth functions and compositions of convex functions with H\"older smooth maps. We first establish the convergence of the stochastic subgradient method and the stochastic proximal subgradient method for unconstrained and composite paraconvex optimization, respectively. The analysis is then extended to a broad family of algorithms that access the objective through stochastic models satisfying certain approximation-quality and continuity assumptions. The main principle that underlies the convergence analysis is that the algorithms under consideration can be interpreted as perturbed descent methods on the generalized Moreau envelope of the original paraconvex problem.
We propose a projected quasi-subgradient method for constrained, nondifferentiable, quasi-convex multiobjective minimization problems. Unlike existing approaches that rely on Lipschitz continuity, our method only requires Hölder continuity of the objective components, thereby covering a broader class of quasi-convex fu...
F. Babu, Debdas Ghosh, Muzaffar Sarkar Raju· Positivity (Dordrecht)· 0 citations
We investigate the optimization problem of minimizing a nonsmooth function that satisfies a nonsmooth version of the descent lemma over a nonempty and closed but not necessarily convex set. The objective function belongs to the class of upper-$\mathcal{C}^2$ functions, whereas the constraints may promote a sparse or lo...
Christian Kanzow, Jannis Krüger, Leo Lehmann· 0 citations
We introduce the notion of a zero-scale asymptotic function. In contrast to the usual asymptotic function, which is related to the slopes of a function at infinity along a given direction, the new function is related to the jumps of the function along that direction. Applications are given to the unconstrained and the...
F. Flores-Bazán, N. Hadjisavvas· Journal of Convex Analysis· 0 citations
We introduce Envelopt, a globally convergent iterative framework for a broad class of structured optimization problems where a smooth objective is augmented by a nonsmooth convex regularizer composed with a smooth mapping, and the variables are subject to general smooth constraints. All smooth functions may be nonconve...
We study nonsmooth convex-concave saddle-point problems over compact convex sets, assuming access to stochastic subgradients of the payoff function. We develop single-loop projection-free algorithms that use linear minimization oracles over the primal and dual domains. Unlike prior projection-free approaches that rely...
This work analyzes a stochastic Newton optimization scheme for locating the unique global minimizer of a general nonconvex objective function and proves global almost sure convergence.
Edward Huynh, Bjorn Engquist· 0 citations
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